AMC 10 · 2004 · #19
Grade 10 geometry-3dPick an answer.
Tool #17 (Visualize Spatial Relationships) is the whole game here: slice the solid with a vertical plane through its axis. Because the sphere's center lies on that axis, the slice cuts the sphere in a great circle of the same radius r, and it cuts the solid in an isosceles trapezoid. Every tangency survives the slice, so the 3D question becomes the flat question 'a circle inscribed in a trapezoid'. Tool #1 (Draw a Diagram) then labels that trapezoid with real numbers. Tool #4 (Introduce a Variable) names the missing height h, and Tool #7 (Identify Subproblems) splits the work into two easy pieces: find the slant side from tangent lengths, then find h from a right triangle.
Slice the solid down its axis
Slicing down the axis turns the solid into a trapezoid with an inscribed circle.
One slice through the axis keeps every tangency and trades a hard 3D picture for a flat trapezoid with a circle squeezed inside.
10.G-MG.A.1Visualize Spatial RelationshipsTouching both bases sets 2r=h
Touching both parallel sides makes the height exactly two radii.
A ball wedged between two parallel floors is exactly as tall as the gap.
10.G-CO.A.1Introduce A VariableTangent lengths give the slant side
Equal tangent lengths make the slanted side 20.
Both tangent lines drawn from one corner reach the circle at the same distance, so the slant side is just the two base radii added together.
Both tangent lines from one corner reach the circle equally far, so the slant side is the two base radii added.
▸ Why?
A tangent meets the radius at the touch point square on, making each tangent a leg of a right triangle.
▸ Why?
Those two right triangles share a hypotenuse and a radius, so their remaining legs are forced to be equal.
Pythagoras on the slanted side
A right triangle then gives the height as 12.
The slant side is the hypotenuse of a right triangle whose horizontal leg is the difference of the radii.
8.G.B.7Draw A DiagramCheck the figure is consistent
Opposite side sums agree, so the figure is consistent.
For a circle to touch all four sides, the two pairs of opposite sides must have the same total length, and here they do.
6.EE.B.6Identify SubproblemsHalve the height
Halving the height gives the radius 6, choice (A).
The height is two radii tall, so halving it hands over the radius.
6.EE.B.6Identify SubproblemsSlice the solid straight down its axis: the ball becomes a circle squeezed inside a trapezoid, equal tangent pieces make the slanted side 18+2=20, Pythagoras turns that into height 12, and the ball's radius is half of it, 6.
- Slice the solid down its axis
- Touching both bases sets 2r=h
- Tangent lengths give the slant side
- Pythagoras on the slanted side
- Check the figure is consistent
- Halve the height